Abstract
Abstract Let $$\begin{array}{} \displaystyle Lf(x)=-\frac{1}{\omega(x)}\sum_{i,j}^{}\partial_{i}(a_{ij}(\cdot)\partial_{j}f)(x)+V(x)f(x) \end{array}$$ be the degenerate Schrödinger operator, where ω is a weight from the Muckenhoupt class A2, V is a nonnegative potential that belongs to a certain reverse Hölder class with respect to the measure ω(x)dx. For such an operator we define the area integral $\begin{array}{} \displaystyle S^{L}_h \end{array}$ associated with the heat semigroup and obtain the area integral characterization of $\begin{array}{} \displaystyle H^{1}_{L} \end{array}$, which is the Hardy space associated with L.
Highlights
As a suitable substitute of Lebesgue spaces Lp(Rn), the classical Hardy space H (Rn) plays an important role in various elds of analysis and partial di erential equations
Let ∆ be the Laplace operator on Rn. It follows from [1] that H (Rn) can be characterized by the maximal function supt> |e−t∆ f (x)|
The Hardy spaces associated with L become one of the most concerned problems of the harmonic analysis
Summary
As a suitable substitute of Lebesgue spaces Lp(Rn), the classical Hardy space H (Rn) plays an important role in various elds of analysis and partial di erential equations. In [11], Fe erman and Stein obtain the area integral characterization of the classical Hardy spaces Hp(Rn). Such characterization was extended to other settings. Let L be a degenerate Schrödinger operator L on Rn. In this paper, motivated by the above literatures, we will prove that the Hardy space associated with L has such a characterization. Let SLh be the area integral associated with the heat semigroup generated by L, see (3.1) below. Since (BMOL(dμ))* is a subclass of the Schwartz temperate distribution space S , so we know that our reproducing formula is valid for the elements in (BMOL(dμ))* due to the fact that for a general potential V, the kernel of e−tL only satis es some Lipschitz condition, see Proposition 3.3.
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